18.2 Endowment budget lines

Let’s consider an agent named Bob. (We’re going to pair him with another agent named Alison in Part VI. Apologies for the multitude of A/B names for agents, it’s a time-honored tradition.)

Review: budget line for an exogenous income

Part II, we might have assumed that Bob would start out with a certain amount of money – say, $m = 120$ dollars – and then buy two goods with that money. For example, suppose $p_1 = 10$ dollars per unit of good 1 and $p_2 = 5$ dollars per unit of good 2; then if he spent all his money on good 1, he could afford \(x_1^\text{max} = {m \over p_1} = {120 \text{ dollars} \over 10 \text{ dollars per unit of good 1}} = 12 \text{ units of good 1}\) Likewise, if he spent all his money on good 2, he could afford \(x_2^\text{max} = {m \over p_2} = {120 \text{ dollars} \over 5 \text{ dollars per unit of good 2}} = 24 \text{ units of good 2}\) His budget line therefore extends from $(12,0)$ to $(0,24)$, with a slope of $-2$ units of good 2 per unit of good 1:

This slope represents the opportunity cost of buying another unit of good 1, in terms of good 2: that is, if he spends $€10$ on another unit of good 1, he has to give up 2 units of good 2 (which would cost $€5$ each). Mathematically, the magnitude of this slope is the price ratio: \(|\text{slope of budget line}| = {p_1 \text{ €/unit of good 1} \over p_2 \text{ €/unit of good 2}} = {p_1 \over p_2} {\text{units of good 2} \over \text{units of good 1}}\) which in this case is $p_1/p_2 = 10/5 = 2$.

Buying and selling from an endowment

Now suppose that instead of Bob has endowment of 8 units of good 1 and 8 units of good 2: that is, he’s starting a consumer optimization problem from a bundle within good 1 - good 2 space. We’ll call this point $E$ for “endowment,” and denote the quantities of goods 1 and 2 in that bundle as $e_1$ and $e_2$.

Let’s assume that Bob can buy or sell goods 1 and 2 at market prices. For example, suppose again that the price of good 1 is $p_1 = 10$ and the price of good 2 is $p_2 = 5$. If Bob wanted to consume more than $e_2 = 8$ units of good 2, he could sell some of his good 1 to buy more good 2, as shown below:

Geometry of the endowment budget line

As in the case with exogenous income, we can draw Bob’s budget line; you can add it to the diagram above by checking the box. How do we get the equation of this budget line?

Using the logic from the example above, we can say that if Bob sells some amount $\Delta x_1$ of good 1, he will earn $p_1 \times \Delta x_1$ from the sale; if he uses the proceeds to buy good 2, the amount of good 2 he can buy is therefore \(\Delta x_2 = \frac{p_1 \times \Delta x_1}{p_2}\) Since $\Delta x_1 = e_1 - x_1$ and $\Delta x_2 = x_2 - e_2$, we can write this equation as \((x_2 - e_2) = \frac{p_1 \times (e_1 - x_1)}{p_2}\) Collecting the $x$ terms on the left-hand side and the $e$ terms on the right hand side, we can write this as \(p_1x_1 + p_2x_2 = p_1e_1 + p_2e_2\) If we compare this to the budget line with income from above, we can see that the left-hand side is the same. We can interpret the right-hand side as the “monetary value” (or “liquidation value”) of the endowment: \(\hat m = p_1e_1 + p_2e_2\) In other words, one way of thinking about the endowment budget constraint is that Bob could sell all his endowment for $\hat m$ dollars, and then go back and spend the money on goods 1 and 2 as usual.

How can we interpret the endpoints of the budget line? Mathematically they’re given by \(\begin{aligned}x_1^\text{max} &= {\hat m \over p_1} = e_1 + {p_2e_2 \over p_1}\\ \\ x_2^\text{max} &= {\hat m \over p_2} = e_2 + {p_1e_1 \over p_2} \end{aligned}\) In the context of trading from an endowment, each of these represents the total amount of a good Bob could afford if he sold all his other goods and used the proceeds to buy that good. For example, the first expression for the maximum amount of good 1 says that if Bob sold all his good 2, he could get $p_2e_2$ for it, and use the proceeds to buy $p_2e_2/p_1$ units of good 1; so his final consumption would be his initial endowment $e_1$ plus $(p_2/p_1)e_2$.

Effect of changes in prices

In the budget line with exogenous income, an increase in the price of either good would lead to a reduction in the size of the budget set, and a decrease would lead to an increase in the size of the budget set. This is because income was fixed, and didn’t change due to a change in prices. With an endowment, the situation is a little more complicated.

We just saw that the monetary value of Bob’s endowment was $p_1e_1 + p_2e_2$. An increase in the price of a good therefore leads to an increase in the monetary value of Bob’s endowment; to the extent that Bob has a lot of good 1 which he might want to sell, this is good for him. However, it also means that buying additional good 1 will be more expensive than it was before. It’s also possible that Bob doesn’t own any good 1, in the which case this is unquestionably bad for him.

Geometrically, one way of thinking about the endowment budget line is that it’s a line that passes through the endowment point and has a slope equal to the price ratio. Therefore:

• an increase in the price ratio (an increase in $p_1$ and/or a decrease in $p_2$) results in a clockwise pivot of the budget line around the endowment point, a lower horizontal intercept, and a higher vertical intercept
• a decrease in the price ratio (a decrease in $p_1$ and/or an increase in $p_2$) results in a counter-clockwise pivot of the budget line around the endowment point, a higher horizontal intercept, and a lower vertical intercept.

The graph below summarizes all of this. The diagram in its initial state shows the situation with $p_1 = 10$ and $p_1 = 5$. If you change the prices, a green dotted line shows this original budget line, so you can see the effect of the change:

A few things to try:

• Double the price of good 1, so $p_1 = 20$ and $p_2 = 5$. In the left-hand graph the budget set shrinks, while in the right-hand graph there’s an area of newly affordable bundles above and to the left of the endowment and an area of newly unaffordable bundles below and to the right. That’s because good 1 is worth more if you sell it, but costs more if you buy it.
• Now double the price of good 2 as well, so $p_1 = 20$ and $p_2 =10$. Relative to the original budget line, the left-hand budget line has a parallel shift in, but the right-hand budget line is unchanged. That’s because the value of the endowment doubles when prices go up, leaving the budget set unchanged.